Graph Homomorphism Convolution

TL;DR

This paper introduces a graph classification method based on graph homomorphism numbers, providing a universal, invariant embedding that is efficient for graphs with bounded tree-width.

Contribution

It proposes a novel graph classification approach using homomorphism vectors, demonstrating universality and efficiency for certain graph families.

Findings

Homomorphism numbers serve as natural graph invariants for classification.

The method is universal in approximating invariant functions.

Efficient for graphs with bounded tree-width.

Abstract

In this paper, we study the graph classification problem from the graph homomorphism perspective. We consider the homomorphisms from $F$ to $G$, where $G$ is a graph of interest (e.g. molecules or social networks) and $F$ belongs to some family of graphs (e.g. paths or non-isomorphic trees). We show that graph homomorphism numbers provide a natural invariant (isomorphism invariant and $\mathcal{F}$-invariant) embedding maps which can be used for graph classification. Viewing the expressive power of a graph classifier by the $\mathcal{F}$-indistinguishable concept, we prove the universality property of graph homomorphism vectors in approximating $\mathcal{F}$-invariant functions. In practice, by choosing $\mathcal{F}$ whose elements have bounded tree-width, we show that the homomorphism method is efficient compared with other methods.